Clues from $\mathcal{Q}$--A null test designed for line intensity mapping cross-correlation studies

Estimating the auto power spectrum of cosmological tracers from line-intensity mapping (LIM) data is often limited by instrumental noise, residual foregrounds, and systematics. Cross-power spectra between multiple lines offer a robust alternative, mitigating noise bias and systematics. However, inferring the auto spectrum from cross-correlations relies on two key assumptions: that all tracers are linearly biased with respect to the matter density field, and that they are strongly mutually correlated. In this work, we introduce a new diagnostic statistic, $\mathcal{Q}$, which serves as a data-driven null test of these assumptions. Constructed from combinations of cross-spectra between four distinct spectral lines, $\mathcal{Q}$ identifies regimes where cross-spectrum-based auto-spectrum reconstruction is unbiased. We validate its behavior using both analytic toy models and simulations of LIM observables, including star formation lines ([CII], [NII], [CI],[OIII]) and the 21-cm signal. We explore a range of redshifts and instrumental configurations, incorporating noise from representative surveys. Our results demonstrate that the criterion $\mathcal{Q} \approx 1$ reliably selects the modes where cross-spectrum estimators are valid, while significant deviations are an indicator that the key assumptions have been violated. The $\mathcal{Q}$ diagnostic thus provides a simple yet powerful data-driven consistency check for multi-tracer LIM analyses.